The game of poker (or at least most of the variants) is considered to be computationally unsolvable. However, methods are being developed to at least approximate perfect strategy from the game theory perspective in the heads-up (two player) game, and increasingly good systems are being created for the multi-player or ring game.
Perfect strategy has multiple meanings in this context. From a game-theoretic optimal point of view, a perfect strategy is one that cannot lose to any other player’s strategy; however, optimal strategy can vary in the presence of sub-optimal players who have weaknesses that can be exploited. In this case, perfect strategy would be one that correctly or closely models those weaknesses and takes advantage of them to make a profit.
Some of these systems are based on Bayes theorem, Nash equilibrium, Monte Carlo simulation and Neural networks.
A large amount of the research is being done at the University of Alberta by the GAMES group led by Jonathan Schaeffer who developed Poki and PsOpt.
Four card poker is a relatively new casino game similar to three card poker, invented by Roger Snow and owned by ShuffleMaster [1].
The player makes an ante bet and may also make an ‘Aces Up’ bet.
Five playing cards are dealt to the player who has to make the best four-card hand possible.
The dealer is dealt five cards face down, and one card face up, a total of six cards. He also he has to make the best four-card hand.
After seeing his cards and the dealer’s face-up card, the player can opt to fold, in which case he receives nothing, or play, by betting between one and three times his ante.
The best four-card hands for player and dealer will be compared according to the following ranking (from best worst):
four-of-a-kind
straight flush
three of a kind
flush
straight
two pairs
pair
high card
If the player has three-of-a-kind or better, he will receive a bonus based on the ante wager as follows: three-of-a-kind: 2 to 1, straight flush 20 – 1, four of a kind 25-1.
The Aces Up bet is resolved independently of the dealer’s hand, purely on the rank of the player’s payout. The specific payout depends on the payout in use, with payouts for a pair-of-aces (pays even money on the Aces Up wager) or better.
The dealer has an advantage in having an extra card from which to select the best four, and the fact that if the player folds, he will lose his ante, even if his hand was better than the dealers. The player gets return from the bonus bet payment and from the ability to raise by more than one unit one the hand is good.
Strategy for when to raise and fold is fairly complex, but with optimal play the ante + play bet has a house edge of about 3.36% of the initial bet [2].
Generally speaking, one should try to set the highest two-card hand that you can legally set (that is, the best two-card hand that still leaves a higher five-card hand behind). More specifically, one should expect and “average” hand to be something like a medium-to-high pair behind in the five-card hand and an ace-high in front. Detailed computer analysis has been done to determine ideal strategy, but this requires memorizing large tables. A close approximation can be done with only a few rules of thumb. If you are playing in a casino, you can always ask that your hand be set “house way” if you are in doubt; most house strategies are quite reasonable and can be quite close to optimal strategy.
If you have no pair, no straights, and no flushes, set the second- and third-highest cards in your two-card hand. For example, with K-Q-J-9-7-4-3, play Q-J and K-9-7-4-3. There are a few minor exceptions to this (for example, with A-Q-10-9-5-4-2 it is slightly better to play Q-9 and A-10-5-4-2), but these are rare and don’t affect your win rate much.
If you have nothing but a single pair, set it in your five-card hand and put the two highest remaining cards in your two-card hand. For example, with A-Q-Q-9-6-5-3, play A-9 and Q-Q-6-5-3. There are no exceptions to this rule. This rule and the rule above will cover 90% of the hands you play.
Two pair is the most common case where strategy isn’t obvious. You can either play the high pair behind and small pair in front, or else two pair behind and high cards in front. The smaller your high pair and higher your remaining cards, the more you should be inclined to play two pair behind. If your side cards are small, or your larger pair is large, split the pairs. You should always split pairs if your high pair is aces, and almost always split if your high pair is kings or queens; they are high enough by themselves. With something like J-J-4-4-A-Q-5 you can consider playing A-Q and J-J-4-4-5-, since A-Q in front is not much worse than 4-4, but two pair behind is much better than a single pair of jacks. Jacks and tens might be more inclined to split, because tens in front is much better than A-Q. With pairs as small as 7s and 8s, you might consider playing two pair behind if you can play a king-high or better in front. With 2s and 3s, you might even play as little as a queen-high in front. If you have no side cards higher than a jack, always split pairs, even 2s and 3s. (Most house ways split if there’s a pair of 6s or higher, and split small pairs if there’s no Ace for the low hand.)
Three pair is a very good hand. Always play the highest pair in front, no exceptions. For example, with K-K-7-7-4-4-A, play K-K and 7-7-4-4-A.
If you have three of a kind and nothing else, play three of a kind behind and remaining high cards in front, unless they are aces–always split three aces, playing a pair of aces behind and ace-high in front. Occasionally, you can even split three kings if your remaining side cards are not queen-high (for example, with K-K-K-J-9-7-6, it is slightly better to play K-J and K-K-9-7-6 than to play J-9 and K-K-K-7-6). Most house ways only split three Aces.
If you can play a straight or a flush or both, play whichever straight-or-better five-card hand makes the best two-card hand. For example, with K♠-9♠-8♣-7♠-6♣-5♠-4♠, playing the flush would put 8-6 in front, playing the 9-high straight would put K-4 up front, but the correct play is K-9 and 8-7-6-5-4. Occasionally, you will have a straight or flush with two pair; in that case, play as if it were two pair and ignore the straight or flush. This rule applies even if you can play a straight flush: if a straight or flush makes a better hand in front, play it that way.
With a full house, generally play trips behind and the pair in front. The exception is if the pair is very small and your side cards are very high, for example, with 5-5-5-3-3-A-Q, it might be better to play A-Q with the full house behind. These are rare, though, and you will never be making a big mistake if you never play a full house behind. House ways will always split the full house.
With two sets of trips, play the higher as a pair in front, and the smaller trips behind. For example, with Q-Q-Q-7-7-7-A, play Q-Q and 7-7-7-A-Q. No exceptions.
With four of a kind, play as if it were two pair, but be slightly less inclined to split. For example, with 10-10-10-10-J-5-4, play 10-10 and 10-10-J-5-4; with 3-3-3-3-K-Q-7, play K-Q and 3-3-3-3-7. Most house ways always split the four of a kind.
With three pair and a straight or flush (only possible with the joker), play as three pair (aces in front).
The cases below will probably never happen to you, but just in case:
With four of a kind and a pair, play the pair in front unless it is very small and the four of a kind is very large. For example, with 9-9-9-9-7-7-K, play 7-7 and 9-9-9-9-K, but with Q-Q-Q-Q-3-3-9, you might play Q-Q and Q-Q-3-3-9. House ways always put the quartet in back and the pair in front.
With a full house and a pair, play the higher pair in front and a full house in back.
With four of a kind and trips, split the four to play a pair in front and full house behind. House ways will tend to break the trips.
With all four aces and the joker, play a pair of aces in front and three aces (or a full house) behind UNLESS your back pair is a pair of kings; you get the honor of gloating on this one.
When certain pay schedules are offered by a video poker machine, players using perfect or near perfect strategy can obtain greater than 100% payback over a sufficiently long period of play. These machines are referred to as “full pay” machines. Casinos place full pay machines alongside other machines with pay schedules that offer a negative return, so it is up to the player to properly identify which video poker machines offer the full pay schedules.
Most full pay machines are configured with a pay schedule that is only full pay when the maximum amount of credits is bet. (See the pay schedule tables later in this article for details.)
Deuces Wild
One variation of video poker, called “Deuces Wild”, can be found with pay schedules that offer up to a theoretical return of 100.762%, when played with perfect strategy. It is also available with other pay schedules that have lesser theoretical returns:
Hand
1 credit
2 credits
3 credits
4 credits
5 credits
Natural Royal Flush
300
600
900
1200
4000*
Four Deuces
200
400
600
800
1000
Wild Royal Flush
25
50
75
100
125
Five of a Kind
15
30
45
60
75
Straight Flush
9
18
27
36
45
Four of a Kind
5
10
15
20
25
Full House
3
6
9
12
15
Flush
2
4
6
8
10
Straight
2
4
6
8
10
Three of a Kind
1
2
3
4
5
Theoretical Return
99.679%
99.679%
99.679%
99.679%
100.762%*
*Notice the gap between the payoff for a Natural Royal Flush played with 4 credits vs. one with 5 credits. The payoff schedule for most video poker machines has a gap like this, such that players who do not play with the maximum number of credits at a time are playing with a negative theoretical return.
Double Bonus
Another variation of video poker, called “Double Bonus”, can be found with pay schedules that offer up to a theoretical return of 100.1725%, when played with perfect strategy. It is also available with other pay schedules that have lesser theoretical returns:
Hand
1 credit
2 credits
3 credits
4 credits
5 credits
Royal Flush
250
500
750
1000
4000*
Straight Flush
50
100
150
200
250
Four Aces
160
320
480
640
800
Full House
10
20
30
40
50
Flush
7
14
21
28
35
Straight
5
10
15
20
25
Three of a Kind
3
6
9
12
15
Two Pair
1
2
3
4
5
Jacks or Better
1
2
3
4
5
Theoretical Return
99.1079%
99.1079%
99.1079%
99.1079%
100.1725%*
*Notice the gap between the payoff for a Royal Flush played with 4 credits vs. one with 5 credits. Players who do not play with the maximum number of credits at a time are playing with a negative theoretical return.
Double Double Bonus
Another variation of video poker, called “Double Double Bonus”, can be found with pay schedules that offer up to a theoretical return of 100.067%, when played with perfect strategy. It is also available with other pay schedules that have lesser theoretical returns:
Hand
1 credit
2 credits
3 credits
4 credits
5 credits
Royal Flush
250
500
750
1000
4000*
Straight Flush
50
100
150
200
250
Four Aces w/2, 3, or 4
400
800
1200
1600
2000
Four 2, 3, or 4 w/A-4
160
320
480
640
800
Four Aces
160
320
480
640
800
Four 2, 3, or 4
80
160
240
320
400
Four 5-K
50
100
150
200
250
Full House
10
20
30
40
50
Flush
6
12
18
24
30
Straight
4
8
12
16
20
Three of a Kind
3
6
9
12
15
Two Pair
1
2
3
4
5
Jacks or Better
1
2
3
4
5
Theoretical Return
98.9154%
98.9154%
98.9154%
98.9154%
100.067%*
*Notice the gap between the payoff for a Royal Flush played with 4 credits vs. one with 5 credits. Players who do not play with the maximum number of credits at a time are playing with a negative theoretical return.
Other Full Pay Games
Other kinds of video poker only have positive theoretical returns when the progressive jackpot is high enough. Many establishments advertise with a billboard when the progressive jackpot is high enough.
Locating Full Pay Games
Although full pay video poker machines are found in many “locals” casinos (located off the Strip) in the Las Vegas market (and in a few Reno casinos), most Strip casinos and casinos in other markets only offer video poker pay schedules with a negative theoretical return.
The maximum bet size is kept fairly small on video poker machines with a full pay schedule (one dollar or less), which makes it impractical to win a large amount of money over any reasonable period of time.
A check-raise in poker is a common deceptive play in which a player checks early in a betting round, hoping someone else will open. The player who checked then raises in the same round.
This might be done, for example, when the first player believes that an opponent has an inferior hand and will not call a direct bet, but that he may attempt to bluff, allowing the first player to win more money than he would by betting straightforwardly.
Of course, if no other player chooses to open, the betting will be checked around and the play will fail.
While it is an important part of poker strategy, in some home games and certain small-stakes casino games, this play is not allowed. It is also frequently not allowed in the game of California lowball.
In poker, to bluff is to bet or raise with an inferior hand, or with a hand believed to be inferior. The term is also used as a noun: a bluff is the act of bluffing.
The bluff is an important part of the strategy of any poker game, though it will come into play more in some games than in others. This is because a bluff is intended to represent a strong hand. For example, bluffs are much stronger in pot-limit and no-limit games, because your opponent will have worse pot odds, in addition to the threat of larger bets in future betting rounds if there are any. On the other hand, bluffing is less common in limit Omaha, because it’s often likely that if you don’t have the hand you represent, one of your opponents does.
Strategy
Do not be predictable. If you always bluff in certain situations, your opponents will figure this out and start calling more. On the other hand, if you never bluff, they will figure that out too and stop calling your non-bluff bets, which is a bad thing—even though you might win the hand, you will fail to win the amount of their call. The exact ideal bluffing frequency in each game situation is a complicated exercise in game theory that you will not be able to solve at the table, so you may have to rely on rules of thumb, prior analysis, experience, and intuition.
General guidelines
Bluffs are more successful with fewer people in the pot. Against only one or two opponents, your chances are often good that no one has a hand good enough to call. Against three or more opponents, at least one of them probably does, so bluffing is unlikely to succeed.
Bluff much less in high-low split games—some very weak hands will call hoping for half the pot, and the likelihood of splitting the pot greatly reduces your pot odds in any case. In some games such as limit Omaha high-low, you would not be giving up much advantage if you never bluffed at all.
In games with many betting rounds, bluffs are more often successful in early rounds rather than late ones. Once other players have put a lot of money into the pot, they are less likely to give up (this tendency is based on the false concept of being “pot-committed” and goes beyond the correct strategy of calling more often with higher pot odds. cf. sunk cost fallacy)
Value bet your strong hands, consider bluffing with hands you are almost sure cannot win any other way, and check the ones in between: On the last betting round, if you have a hand that might be good but that is not very strong, you are probably better off checking and then calling a bet by your opponent rather than bluffing. A player with a worse hand will probably not call if you bet, but a check might induce your opponent to bluff, allowing your call to win more money. On the other hand, a player with a better hand than yours will almost certainly call, and may raise, costing you money. You also do not need the protection of a bet.
A raise, and especially a check-raise, as a bluff is more psychologically intimidating than just opening. Of course it also risks more of your money and makes the pot bigger (and therefore more likely to be called), so it must be used with care.
Semi-bluffs
In games with multiple betting rounds, to bluff on one round with an inferior or drawing hand that might become a much better one by chance in a later round is often called a semi-bluff. Semi-bluffs thus afford a player two opportunities to win the pot: everyone may fold, or the player still might win the showdown if called.
For example, a player in a stud poker game with four spade-suited cards showing (but none among their downcards) on the penultimate round might raise, hoping that others believe they have a flush even though they do not. If their bluff fails and they are called, they still might be dealt a spade on the final card and win the showdown (or they might be dealt another non-spade and try their bluff again, in which case it is a pure bluff or stone-cold bluff on the final round rather than a semi-bluff).
Randomizing devices
In performing bluffs, it often helps to have a randomizing device: for example, if your analysis or experience leads you to believe that you should bluff half of the time in a certain situation, use a device such as the color of the last card dealt. Another strategy useful in short-handed games is to give yourself fake outs: if a jack is not a scare card, pretend that every jack is an out for you, even if it is not. This strategy has a mathematical basis in game theory.
Bluff (the game)
Bluff is an ancient predecessor of poker played in the 1800s, where only the cards from 10 to Ace were used, and straights and flushes hadn’t been invented yet.
The game of poker (or at least most of the variants) is considered to be computationally intractable. However, methods are being developed to at least approximate perfect strategy from the game theory perspective in the heads-up (two player) game, and increasingly good systems are being created for the multi-player or ring game. Perfect strategy has multiple meanings in this context. From a game-theoretic optimal point of view, a perfect strategy is a minimax one that cannot expect to lose to any other player’s strategy; however, optimal strategy can vary in the presence of sub-optimal players who have weaknesses that can be exploited. In this case, a perfect strategy would be one that correctly or closely models those weaknesses and takes advantage of them to make a profit. Some of these systems are based on Bayes theorem, Nash equilibrium, Monte Carlo simulation, and Neural networks. A large amount of the research is being done at the University of Alberta by the GAMES group led by Jonathan Schaeffer who developed Poki and PsOpt.
A major part of the skill of live poker games, however, is guessing at the strength of a player’s hand by identifying tells made by other players, while concealing one’s own. This fundamentally differs from games like chess where all information about the game’s current state is public. As a computer would not make any tells, playing against a computer would fundamentally change the nature of the game far more than chess and similar games.
Although you cannot read a computer opponent, playing against computer opponents can still help you sharpen your skills by learning how to count outs and play the percentages. With the advancing technology of artificial intelligence, computer players can be created to incorporate bluffs and other human-like decisions.
There are several excellent books on poker strategy, and this article will only attempt to deal with the basics that must be mastered by the beginner. A list of articles on the material summarized here appears below.
Hand strength
Once a player has mastered the rank of hands, it is more important to realize their relative strength at a poker table. Approximately half the five-card hands in any given game will be less than a pair of twos, but only about one-fourth of five-card hands in the long run will be better than a pair of Aces. A full house is such a good hand that it is far more likely to be the best hand on the table than a 7 high (the lowest possible hand) is to be the lowest hand at a given table.
One mistake made by many beginners is to bet hands that are unlikely to win in the showdown, hoping that they will eventually improve. In the long run, this is a losing strategy against experienced players. For example, in draw poker, any hand less than a pair should generally be folded at the earliest opportunity. In other games, such as Texas Hold ‘Em where only two cards are dealt before the betting round, unmatched combinations of low cards are unlikely to result in a winning hand.
Improvement and pot odds
Athough improvement is possible with virtually every hand, most beginners forget that players with better hands may also improve their hands on the draw, and that in the long run the player with the better hand before a draw is likely to have the better hand after the draw as well. Generally, if you have reason to believe that your opponent has a better hand than you at any given point of a betting round, the appropriate action is to fold. However, if the money in the pot is large compared to the bet required to stay in (the “pot odds”), a call is possible. This is particularly the case when a player is attempting to fill a straight or flush. However, the approximate odds of filling an outside straight on the next draw is about 6-1, and the odds of similarly filling a flush is about 5-1. As such, calling is not recommended if the money that could be won gives a lesser payout on the bet.
One bet made by beginners that rarely pays off is to fill an “inside straight” – a straight with one of the middle cards missing. The odds against filling such a straight on a single draw are roughly 13-1 against and should not be considered unless the pot odds are particularly good.
Bluffing
Beginners, even talented beginners, see bluffing as a way to “buy” the pot. However, bluffing seldom works against poor players (who tend to overestimate the strength of their hand to begin with) and does not work for long against expert players if a player bluffs too frequently. Although all players should bluff occasionally to make their large bets on good hands look less credible, consistent and constant bluffing generally leads to large losses.
In large games it is difficult to bluff because of the high chance that someone has a good hand. It is also more sensible for your opponents to call than fold if the pot is large unless they are sure their hand is dead. Paying $10 to call with $100 in the pot is good value for anyone with more than a 10% chance of winning.
Position play
Generally, players who have to bet first need stronger hands to open the betting than players who bet later. This is because the raw odds of a better hand being on the table increase based on the number of players who have not had the opportunity to bet. For example, in a six-player game of draw poker, it is recommended that a player check in the first betting position unless they have at least a pair of aces. However, the last player to bet (the dealer) may open the betting if no-one else has spoken with as little as a pair of twos. One expert Hold ‘Em player recommends folding if you are the first player to the left of the blind unless you have the strength to raise.
The game of poker (or at least most of the variants) is considered to be computationally unsolvable. However, methods are being developed to at least approximate perfect strategy from the game theory perspective in the heads-up (two player) game, and increasingly good systems are being created for the multi-player or ring game.
